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TMUA 2016 · Paper 2 · Question 20 of 20

TMUA 2016 Paper 2 Question 20

Coordinate geometry — Interior angles of regular polygons · integer solutions. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 2Coordinate geometryInterior angles of regular polygons · integer solutions8 options
Each interior angle of a regular polygon with n sides is 34 of each interior angle of a second regular polygon with m sides.

How many pairs of positive integers n and m are there for which this statement is true?

  1. Anone
  2. B1
  3. C2
  4. D3
  5. E4
  6. F5
  7. G6
  8. Hinfinitely many
Show the answer and worked solution
answer · E
  1. Anone
  2. B1
  3. C2
  4. D3
  5. E4
  6. F5
  7. G6
  8. Hinfinitely many
The interior angle of a regular n-gon is 180(n−2)n, so the condition is n−2n = 34⋅m−2m. Clearing denominators, 4m(n−2) = 3n(m−2), which tidies to mn − 8m + 6n = 0 and hence m = 6n8−n. A polygon needs at least 3 sides, so test n = 3 upwards: n=3 gives 185, not an integer; n=4, 5, 6, 7 give m = 6,  10,  18,  42; and n ≥ 8 makes m undefined or negative. That is 4 pairs.